Initial-Boundary and Nonlocal Boundary Value Problems for Higher Order Nonlinear Hyperbolic Equations with Two Independent Variables
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Boundary value problems in a characteristic rectangle for nonlinear hyperbolic equations of higher order are considered. The concept of strong well–posedness of a boundary value problem is introduced. For initial–boundary value problems there are established: (i) Necessary and sufficient conditions of strong well–posedness; (ii) Unimprovable sufficient conditions of local and global solvability; (iii) Effective sufficient conditions of solvability of two–point, multi–point, periodic and Dirichlet type problems; (iv) Sharp a priori estimates of solutions of ill–posed initial–boundary value problems; (v) Unimprovable conditions guaranteeing unique solvability of ill–posed initial–boundary value problems. For nonlocal boundary value problems there are established: (i) Necessary and sufficient conditions for a linear problem to have the Fredholm property; (ii) Necessary and sufficient conditions of strong well–posedness; (iii) Optimal sufficient conditions of solvability and unique solvability; (iv) Effective sufficient conditions of solvability of periodic and Dirichlet type problems in case, where the righthand side of the equation has arbitrary growth order with respect to some phase variables.